A355635 - OEIS

A355635

Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -8, 22, -24, 9, 1, -16, 93, -238, 256, -96, 1, -32, 386, -2180, 5825, -6500, 2500, 1, -64, 1586, -19184, 117561, -345600, 407700, -162000, 1, -128, 6476, -164864, 2229206, -15585920, 51583084, -64538880, 26471025

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OFFSET

0,5

COMMENTS

Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(n, 0) = 1.

T(n, 1) = -2^(n-1), for n > 0.

T(n, 2) = A000346(n-2), for n > 1.

T(n, 3) = -A025131(n-1), for n > 1.

T(n, 4) = A025133(n-1), for n > 1.

T(n, n) = (-1)^n*A001142(n-1), for n > 0.

T(n+1, n) = (-1)^n*A025134(n).

T(n+2, n) = (-1)^n*A025135(n).

EXAMPLE

The triangle begins:

1, -1;

1, -2, 1;

1, -4, 5, -2;

1, -8, 22, -24, 9;

1, -16, 93, -238, 256, -96;

1, -32, 386, -2180, 5825, -6500, 2500;

...

Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

PROG

(PARI) T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

CROSSREFS

Cf. A000079, A000346, A025131, A025133, A025134, A025135.

Cf. A001142 (right diagonal unsigned).

Sequence in context: A158471 A158472 A198895 * A118686 A355540 A102610

Adjacent sequences: A355632 A355633 A355634 * A355636 A355637 A355638

KEYWORD

sign,tabl

AUTHOR

Thomas Scheuerle, Jul 11 2022

STATUS

approved